New Results on the Interpolation Problem for Continuous-Time Stationary Increments Processes

1984 ◽  
Vol 22 (1) ◽  
pp. 133-142 ◽  
Author(s):  
Michele Pavon
Keyword(s):  
Continuous Time ◽  
Interpolation Problem ◽  
Stationary Increments

scholarly journals Predicting the Supremum: Optimality of ‘Stop at Once or Not at All’

2012 ◽  
Vol 49 (3) ◽  
pp. 806-820
Author(s):  
Pieter C. Allaart
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Analogous Result ◽  
Stopping Time ◽  
Stable Processes ◽  
Finite Variation ◽  
Lévy Measure ◽  
One Dimensional ◽  
Stationary Increments ◽  
Reverse Relationship

Let (Xt)0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (Xt) ‘as close as possible’ to its eventual supremum MT := sup0 ≤ t ≤ TXt, when the reward for stopping at time τ ≤ T is a nonincreasing convex function of MT - Xτ. Under fairly general conditions on the process (Xt), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.

On the supremum distribution of integrated stationary Gaussian processes with negative linear drift

1999 ◽  
Vol 31 (01) ◽  
pp. 135-157 ◽  
Author(s):  
Jinwoo Choe ◽  
Ness B. Shroff
Keyword(s):  
Gaussian Processes ◽  
Extreme Value Theory ◽  
Continuous Time ◽  
Value Theory ◽  
Numerical Examples ◽  
Stationary Increments ◽  
Negative Drift ◽  
Stationary Gaussian Processes ◽  
Continuous Time Processes ◽  
Asymptotic Upper Bound

In this paper we study the supremum distribution of a class of Gaussian processes having stationary increments and negative drift using key results from Extreme Value Theory. We focus on deriving an asymptotic upper bound to the tail of the supremum distribution of such processes. Our bound is valid for both discrete- and continuous-time processes. We discuss the importance of the bound, its applicability to queueing problems, and show numerical examples to illustrate its performance.

scholarly journals Exact overflow asymptotics for queues with many Gaussian inputs

2003 ◽  
Vol 40 (3) ◽  
pp. 704-720 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Michel Mandjes
Keyword(s):  
Brownian Motion ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Time Horizon ◽  
Exact Results ◽  
Service Capacity ◽  
Stationary Increments ◽  
Overflow Probability ◽  
Finite Time Horizon ◽  
Transient Probability

In this paper we consider a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments. After scaling the buffer exceedance threshold and the (constant) service capacity by the number of sources, we present asymptotically exact results for the probability that the buffer threshold is exceeded. We consider both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon. We give detailed results for the practically important cases in which the inputs are fractional Brownian motion processes or integrated Gaussian processes.

scholarly journals Predicting the Supremum: Optimality of ‘Stop at Once or Not at All’

2012 ◽  
Vol 49 (03) ◽  
pp. 806-820
Author(s):  
Pieter C. Allaart
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Analogous Result ◽  
Stopping Time ◽  
Stable Processes ◽  
Finite Variation ◽  
Lévy Measure ◽  
One Dimensional ◽  
Stationary Increments ◽  
Reverse Relationship

Let (X t )0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (X t ) ‘as close as possible’ to its eventual supremum M T := sup0 ≤ t ≤ T X t , when the reward for stopping at time τ ≤ T is a nonincreasing convex function of M T - X τ. Under fairly general conditions on the process (X t ), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.

Continuous-time monotone stochastic recursions and duality

2000 ◽  
Vol 32 (02) ◽  
pp. 426-445 ◽  
Author(s):  
Karl Sigman ◽  
Reade Ryan
Keyword(s):  
Continuous Time ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Risk Process ◽  
Stationary Increments ◽  
The One ◽  
And Diffusion ◽  
Steady State Probabilities ◽  
Two Barriers

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.

Continuous-time monotone stochastic recursions and duality

2000 ◽  
Vol 32 (2) ◽  
pp. 426-445 ◽  
Author(s):  
Karl Sigman ◽  
Reade Ryan
Keyword(s):  
Continuous Time ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Risk Process ◽  
Stationary Increments ◽  
The One ◽  
And Diffusion ◽  
Steady State Probabilities ◽  
Two Barriers

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.

scholarly journals Exact overflow asymptotics for queues with many Gaussian inputs

2003 ◽  
Vol 40 (03) ◽  
pp. 704-720 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Michel Mandjes
Keyword(s):  
Brownian Motion ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Time Horizon ◽  
Exact Results ◽  
Service Capacity ◽  
Stationary Increments ◽  
Overflow Probability ◽  
Finite Time Horizon ◽  
Transient Probability

In this paper we consider a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments. After scaling the buffer exceedance threshold and the (constant) service capacity by the number of sources, we present asymptotically exact results for the probability that the buffer threshold is exceeded. We consider both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon. We give detailed results for the practically important cases in which the inputs are fractional Brownian motion processes or integrated Gaussian processes.

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